Birkhäuser Advanced Texts Basler Lehrbücher Alessandro Fonda Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations BirkhäuserAdvancedTextsBaslerLehrbücher Serieseditors StevenG.Krantz,WashingtonUniversity,St.Louis,USA ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA JanNekováˇr,UniversitéPierreetMarieCurie,Paris,France Moreinformationaboutthisseriesat:http://www.springer.com/series/4842 Alessandro Fonda Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations AlessandroFonda DipartimentodiMatematicaeGeoscienze UniversitaJdegliStudidiTrieste Trieste,Italy ISSN1019-6242 ISSN2296-4894 (electronic) BirkhaRuserAdvancedTextsBaslerLehrbuRcher ISBN978-3-319-47089-4 ISBN978-3-319-47090-0 (eBook) DOI10.1007/978-3-319-47090-0 LibraryofCongressControlNumber:2016958441 ©SpringerInternationalPublishingAG2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Rodica Contents 1 PreliminariesonHilbertSpaces .......................................... 1 1.1 TheHilbertSpaceStructure ......................................... 1 1.2 SomeExamplesofHilbertSpaces................................... 3 1.3 FundamentalProperties.............................................. 6 1.4 Subspaces............................................................. 8 1.5 OrthogonalSubspaces................................................ 10 1.6 TheOrthogonalProjection........................................... 13 1.7 BasisinaHilbertSpace.............................................. 14 1.8 LinearFunctions...................................................... 19 1.9 WeakConvergence................................................... 25 1.10 ConcludingRemarks................................................. 28 2 OperatorsinHilbertSpaces............................................... 31 2.1 FirstDefinitions ...................................................... 31 2.2 TheAdjointOperator ................................................ 33 2.3 ResolventSetandSpectrum......................................... 36 2.4 SelfadjointOperators................................................. 39 2.5 OperatorsinRealHilbertSpaces.................................... 42 2.6 ConcludingRemarks................................................. 45 3 TheSemilinearProblem................................................... 47 3.1 TheMainProblem ................................................... 47 3.2 PropertiesoftheDifferentialOperator.............................. 49 3.3 TheLinearEquation ................................................. 52 3.3.1 TheCase(cid:2)>0 .............................................. 53 3.3.2 TheCase(cid:2)<0 .............................................. 56 3.3.3 Conclusions .................................................. 57 3.4 TheContractionTheorem............................................ 57 3.5 Nonresonance:ExistenceandUniqueness.......................... 59 3.6 EquationsinHilbertSpaces.......................................... 61 3.7 ConcludingRemarks................................................. 68 vii viii Contents 4 TheTopologicalDegree.................................................... 71 4.1 TheBrouwerDegree................................................. 71 4.2 FurtherConsiderationsontheBrouwerDegree .................... 87 4.3 TheLeray–SchauderDegree......................................... 90 4.4 ConcludingRemarks................................................. 98 5 NonresonanceandTopologicalDegree................................... 101 5.1 TheUseofSchauderTheorem ...................................... 101 5.2 LowerandUpperSolutions.......................................... 104 5.3 TheContinuationPrinciple .......................................... 108 5.4 AsymmetricOscillators.............................................. 111 5.5 NonlinearNonresonance............................................. 112 5.6 Non-bilateralConditions............................................. 119 5.7 TheAmbrosetti–ProdiProblem ..................................... 129 5.8 ConcludingRemarks................................................. 133 6 PlayingAroundResonance................................................ 137 6.1 SomeUsefulInequalities ............................................ 137 6.2 ResonanceattheFirstEigenvalue................................... 139 6.3 Landesman–Lazer:ResonanceatHigherEigenvalues ............. 141 6.4 TheLazer–LeachCondition......................................... 144 6.5 Landesman–LazerConditions:TheAsymmetricCase............. 145 6.6 Lazer–LeachConditionsfortheAsymmetricOscillator........... 149 6.7 MoreSubtleNonresonanceConditions............................. 151 6.8 ConcludingRemarks................................................. 155 7 TheVariationalMethod ................................................... 157 7.1 DefinitionoftheFunctional ......................................... 157 7.2 Minimization ......................................................... 161 7.3 TheEkelandPrinciple................................................ 164 7.4 TheSearchofSaddlePoints......................................... 165 7.5 ConcludingRemarks................................................. 171 8 AtResonance,Again ....................................................... 173 8.1 ResonanceattheFirstEigenvalue................................... 174 8.2 SubharmonicSolutions .............................................. 176 8.3 Ahmad–Lazer–Paul:ResonanceatHigherEigenvalues............ 181 8.4 Landesman–LazervsAhmad–Lazer–Paul.......................... 184 8.5 PeriodicNonlinearities............................................... 187 8.6 ConcludingRemarks................................................. 190 9 Lusternik–SchnirelmannTheory......................................... 193 9.1 ThePeriodicProblemforSystems .................................. 193 9.2 AnEquivalentFunctional............................................ 194 9.3 SomeHintsonDifferentialEquations............................... 197 9.4 Lusternik–SchnirelmannCategory.................................. 199 9.5 MultiplicityofCriticalPoints ....................................... 201 Contents ix 9.6 RelativeCategory .................................................... 206 9.7 ConcludingRemarks................................................. 211 10 ThePoincaré–BirkhoffTheorem ......................................... 213 10.1 TheMultiplicityResult .............................................. 214 10.2 AModifiedSystem................................................... 215 10.3 TheVariationalSetting............................................... 218 10.4 FiniteDimensionalReduction....................................... 221 10.5 PeriodicSolutionsoftheOriginalSystem.......................... 223 10.6 ThePoincaré–BirkhoffTheoremonanAnnulus ................... 225 10.7 ConcludingRemarks................................................. 227 11 AMyriadofPeriodicSolutions ........................................... 231 11.1 EquationsDependingonaParameter............................... 231 11.2 SuperlinearProblems ................................................ 243 11.3 ForcedSuperlinearEquations ....................................... 250 11.4 ConcludingRemarks................................................. 253 A SpacesofContinuousFunctions.......................................... 255 A.1 UniformConvergence................................................ 255 A.2 ContinuousFunctionswithCompactDomains..................... 257 A.3 UniformlyContinuousFunctions.................................... 258 A.4 TheAscoli–ArzelàTheorem......................................... 259 A.5 TheStone–WeierstrassTheorem .................................... 261 B DifferentialCalculusinNormedSpaces ................................. 265 B.1 TheFréchetDifferential.............................................. 265 B.2 SomeComputationalRules.......................................... 267 B.3 TheMeanValueTheorem ........................................... 270 B.4 TheGateauxDifferential............................................. 272 B.5 PartialDifferentials................................................... 273 B.6 TheImplicitFunctionTheorem ..................................... 276 B.7 HigherOrderDifferentials........................................... 282 C ABriefAccountonDifferentialForms .................................. 287 C.1 PreliminaryDefinitions .............................................. 287 C.2 TheExternalDifferential ............................................ 289 C.3 Pull-BackFunctions.................................................. 290 C.4 IntegratingM-DifferentialFormsOverM-Surfaces................ 291 C.5 DifferentiableManifolds............................................. 292 C.6 Orientation............................................................ 293 C.7 TheStokes–CartanTheorem......................................... 294 Bibliography...................................................................... 297 Index............................................................................... 307 Introduction This book is an introductionto the problem of the existence of solutions to some typeofsemilinearboundaryvalueproblems.Itarisesfromaseriesofcourseswhich Ihavegiventoundergraduateandgraduatestudentsinthelastfewyears. The aim of the book is to give the possibility to any good student to reach a research level in this field, starting from the basic knowledge of mathematical analysis which is usually acquired before graduation. To this aim, I will develop sometoolswhichcouldbeusedtoattackmanydifferentboundaryvalueproblems, arising from ordinary or partial differential equations. However, I have chosen to deal mainly with the periodic problem for a second-order scalar ordinary differentialequation.Onereasonforthischoiceisthatthisapparentlysimplemodel alreadyshowsso manydifferentaspects, andcanbe approachedbysuchdifferent techniques, that it seems the ideal starting point to the further understanding of more technical boundary value problems. Another reason comes, of course, from itsintrinsicimportanceintheapplications. So,Iwillbeconcernedwithanequationofthetype x00Cg.t;x/D0; (1) where g W R (cid:2)R ! R is a continuous function, which is T-periodic in its first variable.Themainproblemwillbetofindsomeconditionsonthefunctiongwhich guaranteetheexistenceofT-periodicsolutionsofEq.(1). Moregenerally,wewilldealwiththeproblem (cid:2) x00Cg.t;x/D0; .P/ x.0/Dx.T/; x0.0/Dx0.T/; whereg WŒ0;T(cid:3)(cid:2)R! Riscontinuous.Indeed,ifg.t;x/isdefinedonR(cid:2)R,and T-periodicinitsfirstvariable,itiseasytoseethatanysolutionx.t/ofproblem(P) canbeextendedtothewholeRasaT-periodicsolutionofEq.(1). xi